From Förberedande kurs i matematik 1

The angle \displaystyle \pi-v makes the same angle with the negative x-axis as the angle \displaystyle v makes with the positive x-axis and this means that \displaystyle \pi-v is the reflection of \displaystyle v in the y-axis.

Under such reflection, the angle's y-coordinate does not change (but the x-coordinate changes sign) and therefore \displaystyle \sin (\pi-v) = \sin v = a\,.